WORDS 2017: Combinatorics on Words pp 252-261

# Symmetric Dyck Paths and Hooley’s $$\varDelta$$-Function

• José Manuel Rodríguez Caballero
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10432)

## Abstract

Hooley [6] introduced the function
$$\varDelta (n) := \max _{u \in \mathbb {R}}\# \left\{ d | n: \quad u < \log d \leqslant u+1 \right\} ,$$
where $$\log$$ is the natural logarithm. Changing the base of the logarithm from e to an arbitrary real number $$\lambda > 1$$, we define
$$\varDelta _{\lambda }(n) := \max _{u \in \mathbb {R}}\# \left\{ d | n:\quad u < \log _{\lambda } d \leqslant u+1 \right\} .$$
The aim of this paper is to express $$\varDelta _{\lambda }(n)$$ as the height of a symmetric Dyck path defined in terms of the distribution of the divisors of n.

## Keywords

Dyck path Palindrome Hooley’s $$\varDelta$$-function

## Notes

### Acknowledgement

The author thanks S. Brlek, C. Kassel and C. Reutenauer for they valuable comments and suggestions concerning this research. Also, the author want to express his gratitude to H. F. W. Höft for the useful exchanges of information.

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